Schrodinger equation for nuclear motion

This Hamiltonian is used in the Schrodinger equation for nuclear motion, describing the vibrational, rotational, and translational states of the nuclei. Solving the nuclear Schrodinger equation (at least approximately) is necessary for predicting the vibrational spectra of molecules. 

The first term on the left-hand side of equation (10.18) has the form of a Schrodinger equation for nuclear motion, so that we may identify the expansion coefficient Xk Q) as a nuclear wave function for the electronic state k. The second term couples the influence of all the other electronic states to the nuclear motion for a molecule in the electronic state k. 

The Schrodinger equation for nuclear motion contains a Hamiltonian operator Hop,nuc consisting of the nuclear kinetic energy and a potential energy term which is Eeiec(S) of Equation 2.7. Thus... 

We have vibrationally averaged the CAS /daug-cc-pVQZ dipole and quadmpole polarizability tensor radial functions (equation (14)) with two different sets of vibrational wavefunctions j(i )). One was obtained by solving the one-dimensional Schrodinger equation for nuclear motion (equation (16)) with the CAS /daug-cc-pVQZ PEC and the other with an experimental RKR curve [70]. Both potentials provide identical vibrational... 

Within the context of the Bom-Oppenheimer approximation, the potential energy surface may be regarded as a property of an empirical molecular formula. With a defined PES, it is possible to formulate and solve Schrodinger equations for nuclear motion (as opposed to electronic motion)... 

Having solved for the electronic wave functions and energies, we use the electronic energy including nuclear repulsion U as the potential energy in the Schrodinger equation for nuclear motion ... 

In formulating this new methodology one begins by taking the difference of the Schrodinger equations for nuclear motion in the ground and excited electronic states. Tlien, using the Hermitian property of the vibrational Hamiltonian, it is easy to show that [104] ... 

Most of this chapter deals with the electronic Schrddinger equation for diatomic molecules, but this section examines nucleeu" motion in a bound electronic state of a diatomic molecule. From (13.10) and (13.11), the Schrodinger equation for nuclear motion in a diatomic-molecule bound electronic state is... 

Thus the exact Schrodinger equation for nuclear motion may be rewritten in the form %1 + H-El)f = 0 (11)... 

Equation (15) is of principal significance since it is still the exact Schrodinger equation for nuclear motion in an applicable form. 

For an extension of ADF to a case including a laser field described by vector potential, [495] we study the following non-relativistic Schrodinger equation for nuclear motion on the coordinates R... 

After determining an analytical representation of this surface from the individual energies, these authors carried out calculations of the fuU anhannonic vibrational spectrum of HOCl and HCIO by solving the Schrodinger equation for nuclear motion. The HCIO molecule has not yet been experimentally observed, but these calculations predict that the lowest three vibrational levels of this species lie below its dissociation threshold, so it should be detectable. 

Worth GA, Robb MA, Lasorne B (2008) Solving the time-dependent Schrodinger equation for nuclear motion in one step direct dynamics of non-adiabatic systems. Mol Phys 106 2077... 

Solutions to the Schrodinger equation Hcj) = E(f> are the molecular wave functions 0, that describe the entangled motion of the three particles such that (j) 4> represents the density of protons and electron as a joint probability without any suggestion of structure. Any other molecular problem, irrespective of complexity can also be developed to this point. No further progress is possible unless electronic and nuclear variables are separated via the adiabatic simplification. In the case of Hj that means clamping the nuclei at a distance R apart to generate a Schrodinger equation for electronic motion only, in atomic units,... 

The reader is assumed to be familiar with some of the basic concepts of quantum mechanics. At this point we will therefore just briefly consider a few central concepts, including the time-dependent Schrodinger equation for nuclear dynamics. This equation allows us to focus on the nuclear motion associated with a chemical reaction. 

In chapter 2 we discussed at length the separation of nuclear and electronic coordinates in the solution of the Schrodinger equation. We described the Born-Oppenheimer approximation which allows us to solve the Schrodinger equation for the motion of the electrons in the electrostatic field produced by fixed nuclear charges. There are certain situations, particularly with polyatomic molecules, when the separation of nuclear and electronic motions cannot be made satisfactorily, but with most diatomic molecules the Born-Oppenheimer separation is acceptable. The discussion of molecular electronic wave functions presented in this chapter is therefore based upon the Born-Oppenheimer approximation. 

This can be used to rewrite the diabatic nuclear motion Schrodinger equation for an incomplete set of n electronic states as... 

This last equation is the nuclear Schrodinger equation describing the motion of nuclei. The electronic energy computed from solving the electronic Schrodinger equation (3) on page 163 plus the nuclear-nuclear interactions Vjjjj(R,R) provide a potential for nuclear motion, a Potential Energy Surface (PES). 

Our treatment of the nuclear Schrodinger equation for diatomic molecules has shown that the wave function for nuclear motion can be separated into rotational, vibrational, and translational wave functions ... 

We now consider the nuclear motions of polyatomic molecules. We are using the Born-Oppenheimer approximation, writing the Hamiltonian HN for nuclear motion as the sum of the nuclear kinetic-energy TN and a potential-energy term V derived from solving the electronic Schrodinger equation. We then solve the nuclear Schrodinger equation... 

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